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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1903 edition. Excerpt: ...is convergent. 292. To shew that the expansion of (1 +x)n by the Binomial Theorem is convergent w/ien x 1. Let ur, ur+l represent the rth and (r + l)th terms of the expansion; then u r r "When rn+, this ratio is negative; that is, from this point the terms are alternately positive and negative when x is positive, and always of the same sign when x is negative. Now when r is infinite, Lim----= x numerically; therefore since x 1 the series is convergent if all the terms are of the same sign; and therefore a fortiori it is convergent when some of the terms are positive and some negative. Art. 283. 293. To shew that the expansion of ax in ascending powers ofyiis convergent for every value q/x. Here----=--; and therefore lam----1 whatever be the value of x; hence the series is convergent. 294. To shew that the expansion of log(l + x) in ascending powers of x is convergent when x is numerically less than 1. Here the numerical value of---=----x, which in the limit u, n n--1 is equal to x; hence the series is convergent when x is less than 1. If x=, the serf vergent. Art. 280. If x =--l, the series becomes--1--jr--Q-t--, and is divergent. Art. 290. This shews that the logarithm of zero is infinite and negative, as is otherwise evident from the equation e-"=0. 295. The results of the two following examples are important, and will be required in the course of the present chapter. Example 1. Find the limit of--=--when x is infinite. Put x = e; then logs = y__ y x e y'2 y3 1+y+i2+rs+ 1, y y' also when x is infinite y is infinite; hence the value of the fraction is zero. Example 2. Shew that when n is infinite the limit of nx"=0, when srl. Let x=, so that y 1; 2/ also let?/" = z, so that n log 2/=log z; then m.n=n = l log2 =...show more